Droplet spreading is ubiquitous and plays a significant role in liquid-based energy systems, thermal management devices and microfluidics. While the spreading of non-volatile droplets is quantitatively understood, the spreading and flow transition in volatile droplets remains elusive due to the complexity added by interfacial phase change and non-equilibrium thermal transport. Here we show, using both mathematical modelling and experiments, that the wetting dynamics of volatile droplets can be scaled by the spatial–temporal interplay between capillary, evaporation and thermal Marangoni effects. We elucidate and quantify these complex interactions using phase diagrams based on systematic theoretical and experimental investigations. A spreading law of evaporative droplets is derived by extending Tanner's law (valid for non-volatile liquids) to a full range of liquids with saturation vapour pressure spanning from $10^1$ to $10^4$ Pa and on substrates with thermal conductivity from $10^{-1}$ to $10^3\ {\rm W}\ {\rm m}^{-1}\ {\rm K}^{-1}$. In addition to its importance in fluid-based industries, the conclusions also enable a unifying explanation to a series of individual works including the criterion of flow reversal and the state of dynamic wetting, making it possible to control liquid transport in diverse application scenarios.

The linear stability of the axisymmetric steady flow in a thermocapillary liquid bridge made from 2-cSt silicone oil $({Pr}=28)$ is investigated numerically. The liquid bridge is heated either from above or below and exposed to an axial air flow which is confined to a concentric tube surrounding the bridge. At the annular inlet, the air flow is fully developed and has the same temperature as the adjacent support rod. Using an extended Oberbeck–Boussinesq approximation in which the density of both fluids depends linearly on the temperature in all equations, critical thermocapillary Reynolds numbers are obtained depending on the strength of the imposed axial air flow. The critical conditions are sensitive with respect to the direction of a weak air flow, because the air flow changes the plateau value of the interfacial temperature midway between the hot and cold ends. For stronger air flow the critical thermocapillary Reynolds number almost saturates at moderate values. Throughout, the instability arises as a hydrothermal wave with the gas phase being passive. The dynamic interface deformations for axisymmetric flow caused by the thermocapillarity flow in the liquid and by the stresses from the air flow are considered separately. Apart from turning points of the critical curve, the impact of dynamic surface deformations on the critical thermocapillary Reynolds number is moderate.

The axisymmetric steady two-phase flow of a differentially heated thermocapillary liquid bridge in air and its linear stability is investigated numerically, taking into account dynamic interfacial deformations in the basic flow. Since most experiments require a high temperature difference to drive the flow into the three-dimensional regime, the temperature dependence of the material properties must be taken into account. Three different models are investigated for a high-Prandtl-number thermocapillary liquid bridge with nominal Prandtl number ${\textit {Pr}}=28.8$: the Oberbeck–Boussinesq (OB) approximation, a linear temperature dependence of all material properties and a full nonlinear temperature dependence of all material properties. For all models, critical Reynolds numbers are computed as functions of the volume of the liquid bridge, its aspect ratio, its dimensional size and as a function of the strength of a forced axial flow in the ambient air. Under most circumstances the OB approximation overpredicts and the linear model underpredicts the critical Reynolds number, compared with the model based on the full temperature dependence of the material properties. Among the main influence factors are the proper selection of the reference temperature and, at larger temperature differences, the temperature dependence of the viscosity of the liquid.

The dynamics of the interaction of a system of two thin volatile liquid droplets resting on a soft viscoelastic solid substrate are investigated theoretically. The developed model fully considers the effect of evaporative cooling and the generated Marangoni stresses due to the induced thermal gradients, while also accounting for the effect of the gas phase composition and the diffusion of vapour in the atmosphere of the droplets. Using the framework of lubrication theory, we derive evolution equations for both the droplet profile and the displacement of the elastic solid, which are solved in combination with Laplace's equation for the vapour concentration in the gas phase. A disjoining-pressure/precursor-film approach is used to describe contact-line motion. The evolution equations are solved numerically, using the finite-element method, and we present a thorough parametric analysis to investigate the physical properties and mechanisms that affect the dynamics of droplet interactions. The results show that the droplets interact through both the soft substrate and the gas phase. In the absence of thermocapillary phenomena, the combined effect of non-uniform evaporation due to the increased vapour concentration between the two droplets and elastocapillary phenomena determines whether the drop–drop interaction is attractive or repulsive. The Marangoni stresses suppress droplet attraction at the early stages of the drying process and lead to longer droplet lifetimes. For substrates with intermediate stiffness, the emergence of spontaneous symmetry breaking at late stages of evaporation is found. The rich dynamics of this complex system is explored by constructing a detailed map of the dynamic regimes.

In microfluidic systems, analyte/reagent mixing is essential to achieve rapid chemical/biochemical reactions. The low Reynolds number (Re) flow, however, makes passive mixing difficult and necessitates the use of some external stimulus to cause disturbance in the system. Here, we report how a periodic thermocapillary effect may interact with strategically patterned wall wettability and boost the mixing dynamics in a micro-confined ternary-liquid film system. Approximate, yet without compromise on the physics involved, analytical solutions to the energy and Navier–Stokes equations under creeping flow conditions are obtained to comprehend the thermal and hydrodynamic characteristics of the thermo-capillarity. In the binary fluid limit, our model is validated with the findings of Pendse & Esmaeeli (Intl J. Therm. Sci., vol. 49, 2010, pp. 1147–1155), and good agreement is found. The flow characteristics in the ternary-liquid system due to discrete wall temperatures is also demonstrated via finite-element-based numerical simulations, and agreement with the semi-analytical solution is noted. The qualitative study of the flow pattern reveals that enhanced mixing is obtained as a result of vortical motion created by the interplay of the periodic thermo-capillarity-driven interfacial flow and wall slip. The observation is further consolidated from the numerical simulation of the species transport equation. The species distribution so obtained is compared with fully developed laminar flow for the same inlet concentration. Our study further investigates the effects of the fluids’ relative thermal conductivity, wall slip length, relative film thickness and thermal (or slip) phase differences on the mixing efficiency of the proposed arrangement. While wall slip, relative film thickness and relative thermal conductivity regulate mixing in the top and bottom layers, phase difference determines how well the middle fluid layer mixes. Slip at the walls creates vortex distortions to get a strong churning effect. The lower thermal conductivity of the top and bottom fluids weakens the thermocapillary flow; however, the dominance of the patterned slip improves mixing in such cases.

Thermocapillary droplets with internal thermal singularities have potential applications in drug delivery and cell analysis. Inspired by the work of Pak et al. (J. Fluid Mech., vol. 753, 2014, pp. 535–552), which was investigated for a surfactant-laden non-deformable droplet in an isothermal Poiseuille flow, we have explored the droplet dynamics by taking account of additional internal thermal singularities, namely monopole and dipole. A generalized mathematical model is developed, which is solved by using the solenoidal decomposition to describe the flow field in any arbitrary Stokes flow, and results are shown extensively for the case of a non-isothermal Poiseuille flow. Under small Péclet number ($Pe_s$) limit, the droplet with an off-centred monopole or a dipole oriented along the flow direction shows cross-stream migration at $O(Pe_s^2)$. However, a dipole oriented perpendicular to the flow direction results in an $O(1)$ effect due to thermocapillarity, and from $O(Pe_s)$ onwards, we observe the combined impact of thermocapillary and surfactant-induced Marangoni stresses. As a surprise, we see cross-stream migration of the droplet from the Poiseuille flow centreline in a non-isothermal field, in contrast to existing findings which rule out any cross-stream migration. We show the trade-off between thermal Marangoni number ($Ma_T$) and surfactant Marangoni number ($Ma_\varGamma$). Our findings on droplet dynamics inspire new possibilities for microfluidics-based design.

The dynamics of hydrogen bubbles produced by water electrolysis in an acidic electrolyte is studied using electrochemical and optical methods. A defined cyclic modulation of the electric potential is applied at a microelectrode to produce pairs of interacting H$_2$ bubbles in a controlled manner. Three scenarios of interactions are identified and studied systematically. The most prominent one consists of a sudden reversal in the motion of the first detached bubble, its return to the electrode, and finally its coalescence with the second bubble. Attested by Toepler's schlieren technique, an explanation of contactless motion reversal is provided by the competition between buoyancy and thermocapillary effects.

The surface wave instability (SWI) of thermocapillary migration is examined by linear stability analysis for a droplet on a unidirectional heated plane. Both a Newtonian fluid and an Oldroyd-B fluid are considered. The droplet, flattened by gravity, is susceptible to two kinds of instabilities: convective instability (CI), which is independent of surface deformation; and SWI, which occurs only when the Galileo number and the surface-tension number are not too large. The wavenumber of the latter is much smaller than that of the former, while the reverse is true for the wave speed. SWI is found at different Prandtl numbers (Pr), while its mode includes streamwise and oblique waves. Energy analysis suggests that the energy of the long-wave mode comes from the shear stress induced by the surface deformation, the energy source for the mode with finite wavelength is the work done by Marangoni forces, while the energy from the basic flow is only important in some cases at small Pr. For the Oldroyd-B fluid, a small elasticity slightly changes the critical Marangoni number of SWI, while larger elasticity changes the preferred mode from SWI to CI. The instability mechanism is discussed and comparisons are made with experimental results.

The linear stability of the axisymmetric steady thermocapillary flow in a liquid bridge made from 2 cSt silicone oil (Prandtl number 28) is investigated numerically in the framework of the Boussinesq approximation. The flow and temperature fields in the surrounding gas phase (air) are taken into account for a generic cylindrical container hosting the liquid bridge. The flows in the liquid and in the gas are fully coupled across the hydrostatically deformed liquid–gas interface, neglecting dynamic interface deformations. Originating from a common reference case, the linear stability boundary is computed varying the length of the liquid bridge (aspect ratio), its volume and the gravity level, providing accurate critical data. The qualitative dependence of the critical threshold on these parameters is explained in terms of the characteristics of the critical mode. The heat exchange between the ambient gas and the liquid bridge that is fully resolved has an important influence on the critical conditions.

The influence of thermocapillary stresses on motion of a slowly evaporating single-component droplet in Stokes flow is investigated analytically for the situation where the environment does not have a temperature gradient in the far field. The conservation equations are solved in the liquid and gas phases and coupled at the gas–liquid interface by applying conditions for conservation of mass, species, momentum and energy. It is found that thermocapillary stresses may influence droplet motion by changing the the interface velocity, and that the gas-phase Lewis number of the evaporating component determines whether Marangoni effects increase or decrease droplet drag. If the Lewis number is less than unity, then thermal Marangoni effects increase droplet drag, while if the Lewis number is greater than unity, then thermal Marangoni effects decrease droplet drag. This is related to the sign of the temperature gradient along the droplet surface that is induced by convection. It is found that conditions may exist where a vaporizing droplet in a microgravity environment will exhibit continuous translational motion driven by thermocapillary effects.

The dynamics of a droplet on an inhomogeneously cooled liquid substrate is investigated numerically. The longwave approximation is applied. It is shown that spatial temperature modulation leads to the droplet's motion towards the region of lower temperature, which is accompanied by the change of the droplet shape. An intensive cooling from below can lead to periodic or quasiperiodic oscillations or the droplet's decomposition. A spatial temperature modulation can suppress the oscillatory instability.

Gravity induced film flow over a rigid smoothly corrugated substrate heated uniformly from below, is explored. This is achieved by reducing the governing equations of motion and energy conservation to a manageable form within the mathematical framework of the well-known long-wave approximation; leading to an asymptotic model of reduced dimensionality. A key feature of the approach and to solving the problem of interest, is proof that the leading approximation of the temperature field inside the film must be nonlinear to accurately resolve the thermodynamics beyond the trivial case of ‘a flat film flowing down a planar uniformly heated incline.’ Superior predictions are obtained compared with earlier work and reinforced via a series of corresponding solutions to the full governing equations using a purpose written finite element analogue, enabling comparisons to be made between free-surface disturbance and temperature predictions, as well as the streamline pattern and temperature contours inside the film. In particular, the free-surface temperature is captured extremely well at moderate Prandtl numbers. The stability characteristics of the problem are examined using Floquet theory, with the interaction between the substrate topography and thermo-capillary instability modes investigated as a set of neutral stability curves. Although there are no relevant experimental data currently available for the heated film problem, recent existing predictions and experimental data concerning the behaviour of corresponding isothermal flow cases are taken as a reference point from which to explore the effect of both heating and cooling.

Transient Marangoni convection induced by an isothermal sidewall of a rectangular pool under a zero-gravity condition is studied using scaling analysis. Scaling analysis shows that there exist a number of flow regimes in each evolution scenario, depending on the Marangoni number, the Prandtl number and the aspect ratio. In a typical evolution scenario, a horizontal surface flow and a vertical flow adjacent to the sidewall may appear. Additionally, a number of scaling laws of the velocity and thickness of transient Marangoni convection are obtained. Further, numerical simulation is performed for validation of the selected scaling laws. There exits good agreement between the numerical results and the scaling predictions.

The aim of this paper is to experimentally and analytically study the solutocapillary flow induced in a waterbody due to the presence of a solute source on its surface and the mixing induced by this flow of the solutes and gases dissolved at and near the surface into the waterbody. According to the analytic solution, the induced flow is analogous to a doublet flow in the sense that the flow is directed towards (or away from) the source within a conical region with its vertex at the source, and outside the conical region, the flow is away from (or towards) the source. Our particle image velocimetry and planar laser-induced fluorescence data show that the actual flow is far more complex than the analytic solution because of the influence of other factors which are not accounted for in the analytic model. There is an approximate agreement, but only for the intermediate distances from the source needle. In experiments, the flow changes direction into the waterbody when the solute-induced surface tension gradient driving the flow becomes comparable to the surface tension gradients on the surface due to factors such as contamination and temperature gradients. This causes the mixing of gases dissolved near the surface into the waterbody. Also, as the solute gradient at the surface gives rise to the force that drives the flow, when the solute diffusion coefficient is smaller the flow persists longer because the gradient is maintained for a longer distance.

We investigate the dynamics of a self-rewetting drop placed on a substrate with a constant temperature gradient via three-dimensional numerical simulations using a conservative level-set approach to track the interface of the drop. The surface tension of a so-called self-rewetting fluid exhibits a parabolic dependence on temperature with a well-defined minimum. Two distinct drop behaviours, namely deformation and elongation, are observed when it is placed at the location of the minimum surface tension. The drop spreads slightly and reaches a pseudo-steady state in the deformation regime, while it continuously spreads until breakup in the elongation regime. Theoretical models based on the forces exerted on the drop have been developed to predict the critical condition at which the drop undergoes the transition between the two regimes, and the predictions are in good agreement with the numerical results. We also investigate the effect of the initial position of the drop with respect to the location of the minimum surface tension on the spreading and migration dynamics. It is found that, at early times, the migration of the drop obeys an exponential function with time, but it diverges at the later stage due to an increase in the drop deformation.

The dynamics of thin volatile droplets comprising of binary mixtures deposited on a heated substrate are investigated. Using lubrication theory, we develop a novel one-sided model to predict the spreading and retraction of an evaporating sessile axisymmetric droplet formed of a volatile binary mixture on a substrate with high wettability. A thin droplet with a moving contact line is considered, taking into account the variation of liquid properties with concentration as well as the effects of inertia. The parameter space is explored and the resultant effects on wetting and evaporation are evaluated. Increasing solutal Marangoni stress enhances spreading rates in all cases, approaching those of superspreading liquids. To validate our model, experiments are conducted with binary ethanol–water droplets spreading on hydrophilic glass slides heated from below. The spreading rate is quantified, revealing that preferential evaporation of the more volatile component (ethanol) at the contact line drives superspreading, leading in some cases to a contact line instability. Good qualitative agreement is found between our model and experiments, with quantitative agreement being achieved in terms of spreading rate.

The phase behaviour of fluids at capillary conditions differs from that in bulk media. Therefore, understanding the thermodynamics of solvents in confined media is essential for modelling non-isothermal and non-isobaric engineering applications in porous media – including enhanced oil and heavy-oil recovery. The Thomson equation states that pore sizes have control over the boiling points of liquids in capillary channels. As pore spaces get smaller, boiling temperatures become lower than the normal boiling temperatures of the same liquids. The objective of this paper is to inspect this phenomenon by physically measuring the boiling points of different liquids at capillary conditions and comparing them with the values at bulk conditions and boiling temperatures obtained from the Thomson equation. Several types of microfluidic chips were used as capillary media to observe the phase-change behaviour of heptane, heptane–decane mixture and naphtha. Additionally, vaporization of water, heptane and decane was investigated in Berea sandstone, Indiana limestone, tight sandstone and shale. Pore size distribution analysis was performed to identify the pore diameter variations in each rock sample, and how the existence of extended nanopores in the rocks could impact the phase alteration.

A liquid droplet hovering on a hot solid surface is commonly referred to as a Leidenfrost droplet. In this study, we discover that a Leidenfrost droplet spontaneously performs a series of distinct oscillations as it shrinks during the span of its life. The oscillation first starts out erratically, followed by a stage with stable frequencies, and finally turns into periodic bouncing with signatures of a parametric oscillation and occasional resonances. The last bouncing stage exhibits nearly perfect collisions. We showed experimentally and theoretically the enabling effects of each oscillation mode and how the droplet switches between such modes. We finally show that these spontaneous oscillation modes and the conditions for transitioning between modes are universal for all tested combinations of liquids and surfaces.

Pressure-driven flow in the presence of heat transfer through a microchannel patterned with parallel ridges is considered. The coupled effects of curvature and thermocapillary stress along the menisci are captured. Streamwise and transverse thermocapillary stresses along menisci cause the flow to be three-dimensional, but when the Reynolds number based on the transverse flow is small the streamwise and transverse flows decouple. In this limit, we solve the streamwise flow problem, i.e. that in the direction parallel to the ridges, using a suite of asymptotic limits and techniques – each previously shown to have wide ranges of validity thereby extending results by Hodes et al. (J. Fluid Mech., vol. 814, 2017, pp. 301–324) for a flat meniscus. First, we take the small-ridge-period limit, and then we account for the curvature of the menisci with two further complementary limits: (i) small meniscus curvature using boundary perturbation; (ii) arbitrary meniscus curvature but for small slip (or cavity) fractions using conformal mapping and the Poisson integral formula. Heating and cooling the liquid always degrade and enhance (apparent) slip, respectively, but their effect is greatest for large meniscus protrusions, with positive protrusion (into the liquid) being the most sensitive. For strong enough heating the solutions become complex, suggesting instability, with large positive protrusions transitioning first.

We study the contact line dynamics of a viscous droplet deposited at the centre of a substrate subject to an axial thermal gradient. The temperature of the substrate decreases with distance from the centre, so the Marangoni stress induced at the liquid–air interface displaces the liquid radially outward. The flow experiences two stages. In the first stage, the droplet evolves towards an axially symmetric ring whose radius increases with time as $t^{1/3}$. Using the lubrication approximation, we perform numerical simulations that confirm this law for the motion of the front and show that the maximum thickness of the profile decreases as $t^{-0.374}$. We explain the evolution law of the contact line by balancing Marangoni and viscous stresses. In the second stage, the contact line becomes unstable and develops smooth corrugations whose amplitude increases with time and that eventually become long fingers. The temporal evolution of the Fourier spectra of the contour shows a shift of the most unstable mode from smaller to larger azimuthal wavenumbers.